When one buys a call option I understand that one has the right but not the obligation to buy the shares by option expiration day. If the price of the share has gone past the strike price can you "sell" your call option. Or is the only way to make gain is to actually exercise your "right to buy the shares" and actually buy the shares and than turn around and sell them for the profit?

I have never done a call option but I have read enough about derivatives and it sounds like they're both the same. Warren says that, the industry loves doing derivatives but that is a very risky business because you have two parties with different expectations. One wants the asset to go down and the other might over appreciate the asset they want to sell. Still, investment experts do this type of transactions very often.

There is another transaction called stock arbitrage that basically involves buying a stock of a company that is going to merge for sure. The merger then is willing to pay extra for your shares in that case. Derivatives and Stock Arbitrage are done with borrowed money very often, thereby, increasing the risk of doing such transactions.

The bottom line is, that you should leave this type of transactions, for when you become more experienced, in the art and science of investments.

I looked for the definition of option call and this is what I found . I hope it helps.

Source : Wikipedia

A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option.[1] The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying) from the seller of the option at a certain time (the expiration date) for a certain price (the strike price). The seller (or "writer") is obligated to sell the commodity or financial instrument should the buyer so decide. The buyer pays a fee (called a premium) for this right.

The buyer of a call option purchases it in the hope that the price of the underlying instrument will rise in the future. The seller of the option either expects that it will not, or is willing to give up some of the upside (profit) from a price rise in return for the premium (paid immediately) and retaining the opportunity to make a gain up to the strike price (see below for examples).

Call options are most profitable for the buyer when the underlying instrument moves up, making the price of the underlying instrument closer to, or above, the strike price. The call buyer believes it's likely the price of the underlying asset will rise above the option's strike price by the exercise date. The buyer's maximum loss is limited to the option premium. The profit for the buyer can be very large, and is determined by how high the underlying instrument's spot price rises. When the price of the underlying instrument surpasses the strike price, the option is said to be "in the money".

The call writer does not believe the price of the underlying security is likely to rise above the strike price. The writer sells the call to collect the premium and does not receive any gain if the stock rises above the strike price.

The initial transaction in this context (buying/selling a call option) is not the supplying of a physical or financial asset (the underlying instrument). Rather it is the granting of the right to buy the underlying asset, in exchange for a fee — the option price or premium.
Exact specifications may differ depending on option style. A European call option allows the holder to exercise the option (i.e., to buy) only on the option expiration date. An American call option allows exercise at any time during the life of the option.

Call options can be purchased on many financial instruments other than stock in a corporation. Options can be purchased on futures on interest rates, for example (see interest rate cap), and on commodities like gold or crude oil. A tradeable call option should not be confused with either Incentive stock options or with a warrant. An incentive stock option, the option to buy stock in a particular company, is a right granted by a corporation to a particular person (typically executives) to purchase treasury stock. When an incentive stock option is exercised, new shares are issued. Incentive stock options are not traded on the open market. In contrast, when a call option is exercised, the underlying asset is transferred from one owner to another.
Comment

Why would you want to pay for the right, of buying something your'e not sure it is going to go up, in price anyway ?

Basically a call option is like a bid where you preset a price on a stock but you just don't have any obligation to buy the stock at the strike.
There are obviously 2 scenarios where the stock either rises in value and you lose a part of the profit or the stock dives in value where you make significant profit because you traded in options.